which strikes me as very convoluted (my mother tongue is Italian, but I speak English on a regular basis). What I feel would be natural to say in such circumstances is

"Ich habe dich nicht richtig gehört"

because it seems like a redundancy to specify the way you did not understand something which has been said. On top of that, I was wondering if "verstehen" doesn't itself already imply a deeper understanding of the actual meaning of what's being said. If it did, this would make the expression inexact.
So, what I am wondering here is:

The akustisch is in there exactly because otherwise, as you suggest, verstehen might be understood to refer to the meaning. Nevertheless your Ich habe dich nicht richtig gehört might be preferable.
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Carsten SchultzJan 26 '14 at 16:46

My personal opinion as a native speaker: The accepted answer is correct. But: akustisch nicht verstanden came up as a convoluted way of saying I habe Dich nicht (richtig) gehört towards the late 90s (from what I recall; Google Ngrams puts it much earlier). Some people mixed it up and said Das habe ich jetzt akustisch nicht gehört, which is of course redundant. Sticking with Ich habe Dich nicht richtig gehört is fine.
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RaketenolliMar 19 at 22:44

@ashlotte "akustisch" is very fundamental here, as otherwise people might start explaining what they meant while you actually just wanted them to repeat. There are indeed alternatives, like just explicitly saying that someone should repeat or explaining why you didn't hear them ('you speak too quietly') but the phrase in question has definitely gained most popularity amongst all of them.
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Em1Jan 26 '14 at 21:53

2

@ashlotte... it can be used either way.... "Der Mann war schwer zu verstehen."... meaning "it was hard to hear what he was saying", "Das Thema ist schwer zu verstehen." Meaning... comprehend intelectually. It works both way, thus adding an adverb to specify which way you are referring to is just natural
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EmanuelJan 26 '14 at 22:13

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@ashlotte... wiktionary is usually pretty accurate. As usual best is to check different sources and see how they intersect. Anyway, you question is answered now, is it? (I feel like you're still not convinced)
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EmanuelJan 27 '14 at 0:07

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@ashlotte Well, the point is that it only make sense in spoken language. It doesn't make any sense in written language. For that reason, it's use in restricted to spoken language. That's what they meant.
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Em1Jan 27 '14 at 10:31